It can, but you can't prove that scientifically!
And this this is because Science presupposes certain things, and thus can't explain them, and thus any ultimate explanation of reality is going to be something which science has little to say. Thus to suggest that science is competent to adjudicate the debate over God's existence is like saying that physics can tell us what Moby Dick is about.
What are some of the things science presupposes, and hence can't explain? That there is a physical world external to our minds, is one, mathematics and logic is another. Even laws of nature don't explain anything, for they are merely shorthand descriptions of what things do given the kinds of things they are. They presuppose the existence of things that operate according to them, and hence can't be used to explain why anything exists at all that adheres to them.
Obviously science can't explain morality. Not that it might not say something relevant to morality. But it certainly can't explain why morality has the objective and binding force that it does. (And surely right living is more important than most of what science does say.)
So what to do we have? Science is a valuable resource, a fruitful avenue for gaining knowledge about the physical world. But it cannot justify itself, and leaves much of reality - even physical reality - out of its description of the world, and hence unexplained; for it is not suited to explaining these things. And it can't show that God does not exist. (Though, God's existence can be shown by examining some of the things that science presupposed. And so while God's existence might not scientific, it is hardly anti-scientific or irrational.
And this concludes my meandering remarks on science, at least for now.